Direction of a plane

Question

When we write parametric equations of the plane, we can easily find the direction vector. But for example, if the equation is written like this; x+4y+2z-1=0 we can find the normal vector by coefficients (1,4,2). But how can I find the direction and write the vector equation of the plane ?

Answer 1

Take two vectors perpendicular with the normal vector $\vec {n} $.

for example if $$\vec {n}=(a,b,c) $$ then $$\vec{u}=(-b,a,0) $$ and $$\vec {v}=(-c,0,a) $$ and we are sure they are independent.

or from cartesian equation $$x=1-4y-2z $$ $$y=y $$ $$z=z $$ from here you have the point $(1,0,0) $ and two vectors director. $$\vec {u}=(-4,1,0)$$ $$\vec {v}=(-2,0,1) $$

Answer 2

The normal vector $\vec{n}$ is simply the coefficients of $x,y,z$. Then, if you find a point on the plane, $\vec{p}$, the equation of the plane is $n \cdot \vec{x} = n \cdot \vec{p}$.